# Chaos Theory MS Electrical Engineering Department of Engineering Chaos Theory MS Electrical Engineering Department of Engineering GC University Lahore Course Contents

Introduction Flows on the line Bifurcations Flows on the circle Linear Systems Phase Planes Limit Cycles Lorenz Equation One dimensional Maps

Fractals Strange Attractors Books NONLINEAR DYNAMICS AND CHAOS With Applications to Physics, Biology, Chemistry, and Engineering, STEVEN H. STROGATZ CHAOS AND NONLINEAR DYNAMICS: An Introduction for Scientists and Engineers,

Robert C. Hilborn The illustrated Dictionary of NONLINEAR DYNAMICS AND CHAOS, Tomasz Kapitaniak, Steven R. Bishop Research Chaos, An Interdisciplinary Journal of Nonlinear Science International Journal of Bifurcation and

Chaos Chaos, Solitons and Fractals Chaos Meaning Pronunciation:/kes/noun complete disorder and confusion Example: snow caused chaos in the region Physics: the property of a complex system whose behaviour is

so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. the formless matter supposed to have existed before the creation of the universe.

Reference: Oxford Dictionary Probabilistic vs Deterministic Random Deterministic

Static vs Dynamic Static Systems Dynamic Systems

Dynamics: Subject that deals with change systems that evolve in time, settles down to equilibrium, keeps repeating in cycles, or does something more complicated Brief History of Dynamics 17th Century Newton solving Two-Body Problem (Sun & Earth) using Differential Equations and Law of Gravitation

Three-Body Problem (Sun, Moon and Earth) N o explicit solution Late 19th Century Poincare Qualitative Solution rather than Quantitative, Geometric Approach Dynamics restricted to Nonlinear oscillators in radio, radar, phase-locked loops, and lasers Lorenz's discovery in 1963 of chaotic motion on a strange attractor Weather Model: Aperiodic, very sensitive to Initial Conditions

Types of Dynamical Systems Differential equations: the evolution of systems in continuous time, whereas iterated maps arise in problems Iterated maps (Difference equations): time is discrete

Differential Equations LINEAR Differential Equations Exponential Growth of population of organisms

Differential Equations NONLINEAR Trajectory & Phase-Space Draw the trajectories without actually solving the system Non-autonomous

Systems Time Dependent Systems Dimension of the Phase Space n = 1: Growth, Decay or Equilibrium e.g. RC Circuit (Linear), Logistic Equation (Nonlinear) n = 2: Oscillations e.g RLC Circuit

(Linear), Pendulum (Nonlinear) n >= 3: Three-Body Problem, Chaos & Fractals (Nonlinear) Example: Example (Continued) Application Areas

Mathematics Biology Computer science Economics Engineering Finance

Philosophy Physics Politics Population dynamics Psychology Chaos in Electrical Circuits

Chaos in Electrical Circuits Secure Communication